Hi Manuel -
disturb, bother, and annoy are all synonyms, but in my opinion:
"Do not disturb" - is a polite way of asking/telling
"don't bother me" - firmly telling
"that annoys me" = "that irritates me" ---> this is more intense and stronger than the other two. it's closer to almost being mad about something because it disturbs/bothers to a high degree. hope that helps.